Analytical beam theory for the in-plane deformation of a composite strip with in-plane curvature Anurag Rajagopal ⇑,1 , Dewey H. Hodges 1 Georgia Institute of Technology, Atlanta, GA 30332-0150, United States article info Article history: Available online 18 June 2012 Keywords: Laminated beam Initial curvature Variational asymptotic method Analytical development abstract The variational-asymptotic-method (VAM) provides a mathematically rigorous way to reduce a three- dimensional elasticity formulation to a one-dimensional beam theory without ad hoc assumptions. In this work, the VAM is employed to develop a beam theory to analyze the in-plane deformation of a laminated strip-beam with initial in-plane curvature. The cross-sectional stiffness constants and recovery relations for stress and strain are presented as analytical expressions. For the case of zero initial curvature, consistency of the expressions with those of plate theory is demonstrated. For strip-beams with initial curvature in the in- plane direction, results obtained show explicit dependence on the curvature. Results are verified by com- parison with those obtained from VABS, the accuracy and consistency of which with three-dimensional finite elements has been reported in several published works. In addition to the internal consistency check this work provides and its utility in helping to validate VABS (which is based on the principles of VAM), it is hoped that the results obtained herein, since they are all analytical expressions, will help researchers and engineers validate the effect of initial curvature in their beam theories, whether existing or new. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The advent of composite materials has revolutionized the field of structural engineering, most notably due to their high strength-to-weight ratio and their directional tailorability. Model- ing of structures with one dimension significantly larger than the other two as beams results in a much simpler mathematical for- mulation and helps save computational costs. With the advent of composites and structural members with initial twist/curvature, particularly in the field of aerospace engineering, using beam theories based on traditional approaches/ideas will not yield accu- rate results. Modeling of slender structural members with initial curvature is thus of paramount importance. The VAM provides a rigorous framework to model such structures without ad hoc assumptions regarding their deformation. The deformation is expressed in terms of an unknown set of warping functions, which is extracted using an asymptotic analysis of the variational problem using the system’s inherent small parameters. The computer program VABS (Variational Asymptotic Beam Section) is constructed on the principles of the VAM. One of first significant works concerning VABS was that of Ref. [1]. By the early 2000s, with the work of Ref. [2], VABS was estab- lished as an analysis tool of good standing in the circles of both academia and industry. Recent updates and developments to VABS are discussed in detail in [3,4]. Though novel ideas are not lacking in some of the beam theories in the current literature, the capability and generality of a VAM framework has maintained the superiority of VABS, subsequently making it a popular analysis tool for helicopter blades and wind turbines. Since then several efforts have contributed to the validation and verification of VABS results, and this paper is one such effort. In this work, we propose a beam theory to analyze the in-plane deformation of an initially curved laminated strip-beam. A beam theory must address the following three aspects: a cross-sectional analysis leading to a stiffness matrix which is input into the 1D analysis, the 1D analysis itself, and the formulae or procedure to recover stress, strain and 3D displacement. This paper is organized as follows: Section 2 outlines the theoretical development leading to the results for the first and third aspects described previously. Section 3 demonstrates extraction of some stiffness terms using an equivalent plate theory. Section 4 validates the current work using results from VABS. Finally, conclusions are drawn. 2. Beam theory Consider a laminated strip beam with initial curvature k 3 = 1/R as shown in Fig. 1. This section deals with the development of a 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.06.010 ⇑ Corresponding author. E-mail addresses: r_anurag87@gatech.edu (A. Rajagopal), dhodges@gatech.edu (D.H. Hodges). 1 Daniel Guggenheim School of Aerospace Engineering, United States. Composite Structures 94 (2012) 3793–3798 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct